X-Men movies producer Simon Kinberg has confirmed that there is indeed a connection between X-Men: First Class characters Azazel and Mystique: they're father and daughter (respectively)! Speaking with IGN during the recent X-Men: Days of Future Past Watch Along party, Kinberg dropped the Mystique / Azazel connection during Mystique's (Jennifer Lawrence) scene infiltrating Trask Industries in disguise. That scene is famous for a horrific Easter egg where Mystique opens of mutants that Trask has killed and dissected - including First Class characters Azazel (Jason Fleyming) and Angel (Zoe Kravitz). When Azazel is shown, Kinberg says "There's Azazel... who we know is her dad, but she never gets to learn that."
The X-Men movies connecting the two Marvel Comics characters in such a way is going to be somewhat disturbing for longtime X-Men fans. That's because, in the comics, Azazel and Mystique are actually lovers!
As X-Men lore goes, Azazel is the leader of an ancient race of demonic mutants, who lost a war with an angelic faction of mutants, and was banished to another dimension. His teleporting powers allowed him to briefly return to Earth for stints of time - which Azazel used for tracking down and impregnating uniquely strong and/or gifted women (naturally...). Eventually, he connected with Mystique while the latter was living in Germany disguised as a baron's wife. The child that Azazel and Mystique produce turns out to be none other than the X-Men's Nightcrawler, who has his mom's blue skin, dad's demonic look, and teleporting power, as well as the German background he was raised in.
So, if you thought that Fox's X-Men First Class timeline made it weird by having Professor Xavier and Mystique be childhood friends; or having Magneto seduce a questionably young Raven; you can now add this little family connection to the list. If anything, this will likely be more a point of frustration for fans than a welcome Easter egg; one more example of how Fox's films really did upend X-Men comic book continuity to tell their saga. Fox already confused things just by starting the First Class timeline as some sort of soft reboot, which established character relationships (like Raven and Charles) that in no way matched the original X-Men trilogy timeline. The fact that Azazel and Mystique were related, but got no dramatic time together onscreen, seems like a big missed connection. It only goes to make the later Mystique / Nightcrawler non-connection in X-Men: Apocalypse feel that much weirder.
Thankfully, the X-Men continuity will get a fresh start when it's rebooted into the Marvel Cinematic Universe.